14.06 Midterm
Spring 2004
You have 1.5 hours.
Total points: 100.
luck!
You must answer all questions. No books or notes are allowed.
Answer question 1 in either continuous or discrete time. Good
Question 1 (Continuous Time)
Consider the neoclassical growth model, in continuous time. Assume that there is no population
growth and no exogenous technological change; and normalize the size of the population and labor
to 1. We examine decentralized competitive allocations.
We introduce a government in the model. The only role of the government is to redistribute income (there is no government spending). The government imposes a proportional tax on
household income and uses the collected taxes to finance a lump-sum transfer to each household.
Denote the time-invariant tax rate with τ ∈ [0, 1) and the transfer with Tt .
The budget of the representative household is thus
ct + it = (1 − τ ) (rt kt + wt ) + Tt ,
(1)
where ct denotes consumption, it investment, kt capital, rt the interest rate, and wt the wage rate.
The capital stock accumulates according to
k̇t = it − δkt ,
(2)
where δ ∈ (0, 1). The household’s lifetime utility is given by
U=
Z
∞
e−ρt u(ct )dt,
0
u(c) =
c1−1/θ
,
1 − 1/θ
(3)
where ρ > 0 is the discount rate and θ > 0 is the elasticity of intertemporal substitution. The
household maximizes (3) subject to (1) and (2), taking wt and rt , as well as Tt and τ , as given.
Firms are perfectly competitive and maximize profits. The technology is Cobb-Douglas. Output at time t is thus
yt = f (kt ) = ktα
where kt denotes the capital stock, yt denotes output or income, and α ∈ (0, 1). Note that, in
equilibrium,
yt = rt kt + wt .
1
Finally, the government budget requires that the tax revenue covers the cost of the transfer, so
that
Tt = τ (rt kt + wt ) = τ yt .
(a) Derive the Euler condition that characterizes the optimal consumption behavior of households? How does consumption growth depends on the tax rate τ ? Interpret.
15 points
(b) What is rt in equilibrium? Does a higher kt increase or reduce rt ? Interpret.
points
10
(c) Consider now the general equilibrium. Write down the system of two first-order differential
equations that characterizes the dynamics of consumption and capital. (Remark: These two
conditions are the Euler condition and the resource constraints, expressed in such a way that the
system depends only on c, k, ċ, k̇, and the exogenous parameters, namely τ , θ, α, δ, and ρ.)
10
points
(d) Solve for the steady-state value of capital. How does it depend on τ ? Interpret the effect
of the tax rate on capital accumulation.
10 points
(e) Draw the phase diagram (that is, the ċ = 0 and k̇ = 0 loci together with the saddle path)
for two cases: τ = τ high and τ = τ low , where τ low < τ high . Suppose the economy has been forever
in the steady state with τ = τ high > 0. Suddenly and unexpectedly, at some time t = t0 , the
government reduces the tax to τ low . Following the tax cut, τ is expected to stay at τ low forever.
How does consumption respond at t = t0 ? (Does it rise, fall, or is it ambiguous?) Give the
intuition. (Explain the associated substitution and wealth (or income) effects.) Finally, how does
capital and output evolve over time following the tax cut?
15 points
Question 1 (Discrete Time)
Consider the neoclassical growth model, in discrete time. Assume that there is no population
growth and no exogenous technological change; and normalize the size of the population and
labor to 1. We examine decentralized competitive allocations.
We introduce a government in the model. The only role of the government is to redistribute income (there is no government spending). The government imposes a proportional tax on
household income and uses the collected taxes to finance a lump-sum transfer to each household.
Denote the time-invariant tax rate with τ ∈ [0, 1) and the transfer with Tt .
The budget of the representative household is thus
ct + it = (1 − τ ) (rt kt + wt ) + Tt ,
2
(4)
where ct denotes consumption, it investment, kt capital, rt the interest rate, and wt the wage rate.
The capital stock accumulates according to
kt+1 − kt = it − δkt ,
(5)
where δ ∈ (0, 1). The household’s lifetime utility is given by
U=
∞
X
β t u(ct ),
u(c) =
t=0
c1−1/θ
,
1 − 1/θ
(6)
1
, where ρ > 0 is the discount rate) and θ > 0 is the
where β is the discount factor (β = 1+ρ
elasticity of intertemporal substitution. The household maximizes (6) subject to (4) and (5),
taking wt and rt , as well as Tt and τ , as given.
Firms are perfectly competitive and maximize profits. The technology is Cobb-Douglas. Output at time t is thus
yt = f (kt ) = ktα
where kt denotes the capital stock, yt denotes output or income, and α ∈ (0, 1). Note that, in
equilibrium,
yt = rt kt + wt .
Finally, the government budget requires that the tax revenue covers the cost of the transfer, so
that
Tt = τ (rt kt + wt ) = τ yt .
(a) Derive the Euler condition that characterizes the optimal consumption behavior of households? How does consumption growth depends on the tax rate τ ? Interpret.
15 points
(b) What is rt in equilibrium? Does a higher kt increases or reduces rt ? Interpret.
points
10
(c) Consider now the general equilibrium. Write down the system of two first-order difference
equations that characterizes the dynamics of consumption and capital. (Remark: These two
conditions are the Euler condition and the resource constraints, expressed in such a way that the
system depends only on ct , ct+1 , kt , kt+1 and the exogenous parameters, namely τ , θ, α, δ, and β
or ρ.)
10 points
(d) Solve for the steady-state value of capital. How does it depend on τ ? Interpret the effect
of the tax rate on capital accumulation.
10 points
(e) Now consider the phase diagram, as if time were continuous. Draw the ċ = 0 and k̇ = 0
loci together with the saddle path for two cases: τ = τ high and τ = τ low , where τ low < τ high .
3
Suppose the economy has been forever in the steady state with τ = τ high > 0. Suddenly and
unexpectedly, at some time t = t0 , the government reduces the tax to τ low . Following the tax
cut, τ is expected to stay at τ low forever. How does consumption respond at t = t0 ? (Does it
rise, fall, or is it ambiguous?) Give the intuition. (Explain the associated substitution and wealth
(or income) effects.) Finally, how does capital and output evolve over time following the tax cut?
15 points
Question 2
(a) Can the Solow growth model help us explain the cross-country differences in income levels?
Can it also help us explain the cross-country differences in saving rates?
10 points
(b) In the context of the Ramsey model, can cross-country differences in tax policies help
explain the cross-country differences in income levels and/or saving rates? (Hint: The results of
Question 1 should help you answer this question.)
10 points
(c) Consider the Ramsey model with exogenously fixed labor. Briefly describe the response
of aggregate consumption, investment, and output to a permanent increase in the level of TFP.
Briefly give the intuition.
20 points
Good Luck!
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